1/3+1/[3+5]+1/[3+5+7]+.+1/[3+5+7+.+21]=
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分母an=3+5+7+...+(2n+1)
=(2n+1+3)*n/2=(n+2)n
∴1/an=1/[n(n+2)]=1/2[1/n-1/(n+2)]
∴1/3+1/[3+5]+1/[3+5+7]+.+1/[3+5+7+.+21]
=1/2(1-1/3)+1/2(1/2-1/4)+1/2(1/3-1/5)+.+1/2(1/9-1/11)+1/2(1/10-1/12)
=1/2(1+1/2-1/11-1/12)
=1/2(3/2-23/132)
=1/2*(198-23)/132
=175/264
=(2n+1+3)*n/2=(n+2)n
∴1/an=1/[n(n+2)]=1/2[1/n-1/(n+2)]
∴1/3+1/[3+5]+1/[3+5+7]+.+1/[3+5+7+.+21]
=1/2(1-1/3)+1/2(1/2-1/4)+1/2(1/3-1/5)+.+1/2(1/9-1/11)+1/2(1/10-1/12)
=1/2(1+1/2-1/11-1/12)
=1/2(3/2-23/132)
=1/2*(198-23)/132
=175/264
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